Theorem undifss 3339

Description: Union of complementary parts into whole. (Contributed by Jim Kingdon, 4-Aug-2018.)

Assertion

Ref	Expression
undifss	|- ( A C_ B <-> ( A u. ( B \ A ) ) C_ B )

Proof of Theorem undifss

Step	Hyp	Ref	Expression
1		difss 3108	. . . 4 |- ( B \ A ) C_ B
2	1	jctr 308	. . 3 |- ( A C_ B -> ( A C_ B /\ ( B \ A ) C_ B ) )
3		unss 3156	. . 3 |- ( ( A C_ B /\ ( B \ A ) C_ B ) <-> ( A u. ( B \ A ) ) C_ B )
4	2, 3	sylib 120	. 2 |- ( A C_ B -> ( A u. ( B \ A ) ) C_ B )
5		ssun1 3145	. . 3 |- A C_ ( A u. ( B \ A ) )
6		sstr 3016	. . 3 |- ( ( A C_ ( A u. ( B \ A ) ) /\ ( A u. ( B \ A ) ) C_ B ) -> A C_ B )
7	5, 6	mpan 415	. 2 |- ( ( A u. ( B \ A ) ) C_ B -> A C_ B )
8	4, 7	impbii 124	1 |- ( A C_ B <-> ( A u. ( B \ A ) ) C_ B )

Syntax hints:   /\ wa 102   <-> wb 103   \ cdif 2979   u. cun 2980   C_ wss 2982

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065

This theorem depends on definitions:  df-bi 115  df-tru 1288  df-nf 1391  df-sb 1688  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-v 2612  df-dif 2984  df-un 2986  df-in 2988  df-ss 2995

This theorem is referenced by:  difsnss  3551  exmidundif  3991  undifdcss  6467